electromechanical dampers for vibration control of structures and rotors

The working principle of eddy current dampers is based on the magnetic interaction generated by a magnetic flux linkage’s variation in a conductor Crandall et al., 1968, Meisel, 1984.. I

Electromechanical Dampers for Vibration

Control of Structures and Rotors

Andrea Tonoli, Nicola Amati and Mario Silvagni

Mechanics Department, Mechatronics Laboratory - Politecnico di Torino

Italy

To the memory of Pietro, a model student, a first- class engineer, a hero

1 Introduction

Viscoelastic and fluid film dampers are the main two categories of damping devices used for

the vibration suppression in machines and mechanical structures Although cost effective

and of small size and weight, they are affected by several drawbacks: the need of elaborate tuning to compensate the effects of temperature and frequency, the ageing of the material and their passive nature that does not allow to modify their characteristics with the operating conditions Active or semi-active electro-hydraulic systems have been developed

to allow some forms of online tuning or adaptive behavior More recently, electrorheological, (Ahn et al., 2002), (Vance & Ying, 2000) and magnetorheological (Vance & Ying, 2000) semi-active damping systems have shown attractive potentialities for the adaptation of the damping force to the operating conditions However, electro-hydraulic, electrorheological, and magnetorheological devices cannot avoid some drawbacks related to the ageing of the fluid and to the tuning required for the compensation of the temperature and frequency effects

Electromechanical dampers seem to be a valid alternative to viscoelastic and hydraulic ones due to, among the others: a) the absence of all fatigue and tribology issues motivated by the absence of contact, b) the small sensitivity to the operating conditions, c) the wide possibility

of tuning even during operation, and d) the predictability of the behavior The attractive potentialities of electromechanical damping systems have motivated a considerable research effort during the past decade The target applications range from the field of rotating machines to that of vehicle suspensions

Passive or semi-active eddy current dampers have a simpler architecture compared to active closed loop devices, thanks to the absence of power electronics and position sensors and are intrinsically not affected by instability problems due to the absence of a fast feedback loop The simplified architecture guarantees more reliability and lower cost, but allows less flexibility and adaptability to the operating conditions The working principle of eddy current dampers is based on the magnetic interaction generated by a magnetic flux linkage’s variation in a conductor (Crandall et al., 1968), (Meisel, 1984) Such a variation may be generated using two different strategies:

• moving a conductor in a stationary magnetic field that is variable along the direction of

the motion;

• changing the reluctance of a magnetic circuit whose flux is linked to the conductor

In the first case, the eddy currents in the conductor interact with the magnetic field and

generate Lorenz forces proportional to the relative velocity of the conductor itself In

(Graves et al., 2000) this kind of damper are defined as “motional” or “Lorentz” type In the

second case, the variation of the reluctance of the magnetic circuit produces a time variation

of the magnetic flux The flux variation induces a current in the voltage driven coil and,

therefore, a dissipation of energy This kind of dampers is defined in (Nagaya, 1984) as

“transformer”, or “reluctance” type

The literature on eddy current dampers is mainly focused on the analysis of “motional”

devices Nagaya in (Nagaya, 1984) and (Nagaya & Karube, 1989) introduces an analytical

approach to describe how damping forces can be exploited using monolithic plane

conductors of various shapes Karnopp and Margolis in (Karnopp, 1989) and (Karnopp et

al., 1990) describe how “Lorentz” type eddy current dampers could be adopted as

semi-active shock absorbers in automotive suspensions The application of the same type of eddy

current damper in the field of rotordynamics is described in (Kligerman & Gottlieb, 1998)

and (Kligerman et al., 1998)

Being usually less efficient than “Lorentz” type, “transformer” eddy current dampers are

less common in industrial applications However they may be preferred in some areas for

their flexibility and construction simplicity If driven with a constant voltage they operate in

passive mode while if current driven they become force actuators to be used in active

configurations A promising application of the “transformer” eddy current dampers seems

to be their use in aero-engines as a non rotating damping device in series to a conventional

rolling bearing that is connected to the main frame with a mechanical compliant support

Similarly to a squeeze film damper, the device acts on the non rotating part of the bearing

As it is not rotating, there are no eddy currents in it due to its rotation but just to its

whirling The coupling effects between the whirling motion and the torsional behavior of

the rotor can be considered negligible in balanced rotors (Genta, 2004)

In principle the behaviour of Active Magnetic Dampers (AMDs) is similar to that of Active

Magnetic Bearings (AMBs), with the only difference that the force generated by the actuator

is not aimed to support the rotor but just to supply damping The main advantages are that

in the case of AMDs the actuators are smaller and the system is stable even in open-loop

(Genta et al., 2006),(Genta et al., 2008),(Tonoli et al., 2008) This is true if the mechanical

stiffness in parallel to the electromagnets is large enough to compensate the negative

stiffness induced by the electromagnets

Classical AMDs work according to the following principle: the gap between the rotor and

the stator is measured by means of position sensors and this information is then used by the

controller to regulate the current of the power amplifiers driving the magnet coils

Self-sensing AMDs can be classified as a particular case of magnetic dampers that allows to

achieve the control of the system without the introduction of the position sensors The

information about the position is obtained by exploiting the reversibility of the

electromechanical interaction between the stator and the rotor, which allows to obtain

mechanical variables from electrical ones

The sensorless configuration leads to many advantages during the design phase and during

the practical realization of the device The intrinsic punctual collocation of the not present

sensor avoids the inversion of modal phase from actuator to sensor, with the related loss of

the zero/pole alternation and the consequent problems of stabilization that may affect a

sensed solution Additionally, getting rid of the sensors leads to a reduction of the costs, the

reduction of the cabling and of the overall weight

The aim of the present work is to present the experience of the authors in developing and

testing several electromagnetic damping devices to be used for the vibration control

A brief theoretical background on the basic principles of electromagnetic actuator, based on

a simplified energy approach is provided This allow a better understanding of the

application of the electromagnetic theory to control the vibration of machines and

mechanical structures According to the theory basis, the modelling of the damping devices

is proposed and the evidences of two dedicated test rigs are described

2 Description and modelling of electromechanical dampers

2.1 Electromagnetic actuator basics

Electromagnetic actuators suitable to develop active/semi-active/passive damping efforts

can be classified in two main categories: Maxwell devices and Lorentz devices

For the first, the force is generated due to the variation of the reluctance of the magnetic

circuit that produces a time variation of the magnetic flux linkage In the second, the

damping force derives from the interaction between the eddy currents generated in a

conductor moving in a constant magnetic field

Fig 1 Sketch of a) Maxwell magnetic actuator and b) Lorentz magnetic actuator

For both (Figure 1), the energy stored in the electromagnetic circuit can be expressed by:

(i t( )) flowing in the coil, and the mechanical power is the product of the force (f t( )) and

speed (q t$( )) of the moving part of the actuator

Considering the voltage (v(t)) as the time derivative of the magnetic flux linkage (λ(t)),

eq.(1) can be written as:

( ) ( ) ( ) ( ) ( ) 1 ( )

q t

q

d t

E i t f t q t dt i t d f t dq E E dt

λ

λ λ

In the following steps, the two terms of the energy E will be written in explicit form With

reference to Maxwell Actuator, Figure 1a, the Ampère law is:

a a fe fe

where H a and H fe indicate the magnetic induction in the airgap and in the iron core while l a

and l fe specify the length of the magnetic circuit flux lines in the airgap in the same circuit

The product Ni is the total current linking the magnetic flux (N indicates the number of

turns while i is the current flowing in each wire section) If the magnetic circuit is designed

to avoid saturation into the iron, the magnetic flux density B can be related to magnetic

induction by the following expression:

Considering that (µfe>>µ0) and noting that the total length of the magnetic flux lines in the

airgap is twice q, eq.(3) can be simply written as:

0

2Bq

Ni

The expressions of the magnetic flux linking a single turn and the total number of turns in

the coil are respectively:

airgap

BS

2 0

2

airgap airgap

N S

q

Hence, knowing the expression (eq.(7)) of the total magnetic flux leakage, the Eλof eq (1)

for a generic flux linkage λ and air q, can be computed as:

( )

1

0

2 2

λ

Note that this is the total contribution to the energy (E) if no external active force is applied

to the moving part

Finally, the force generated by the actuator and the current flowing into the coil can be

computed as:

2 2

0 airgap

E f

λμ

∂

2 0

2

airgap

q E

2 2 0

2

4

airgap

N S i f

q

μ

Considering the Lorentz actuator (Figure 1 b), if the coil movement q is driven while the

same coil is in open circuit configuration so that no current flows in the coil, the energy (E) is

zero as both the integrals in eq (1) are null In the case the coil is in a constant position and

the current flow in it varies from zero to a certain value, the contribution of the integral

leading to (E q ) is null as the displacement of the anchor (q) is constant while the integral

leading to ( Eλ) can be computed considering the total flux leakage

0

The first term is the contribution of the magnetic circuit (R is the radius of the coil, q is the

part of the coil in the magnetic field), while the second term is the contribution to the flux of

the current flowing into the coil Current can be obtained from eq.(12) as:

0

i L

Finally computing the derivative with respect to the displacement and to the flux, the force

generated by the actuator and the current flowing into the coil can be computed:

( 0)2

E i

L λ λλ

The equations above mentioned represent the basis to understand the behaviour of

electromagnetic actuators adopted to damp the vibration of structures and machines

2.2 Classification of electromagnetic dampers

Figure 2 shows a sketch representing the application of a Maxwell type and a Lorentz type

actuator In the field of damping systems the former is named transformer damper while the

latter is called motional damper The transformer type dampers can operate in active mode

if current driven or in passive mode if voltage driven The drawings evidence a compliant

supporting device working in parallel to the damper In the specific its role is to support the

weight of the rotor and supply the requested compliance to exploit the performance of the

damper (Genta, 2004) Note that the sketches are referred to an application for rotating

systems The aim in this case is to damp the lateral vibration of the rotating part but the

concept can be extended to any vibrating device In fact, the damper interacts with the non

rotating raceway of the bearing that is subject only to radial vibration motion

2.3 Motional eddy current dampers

The present section is devoted to describe the equations governig the behavior of the

motional eddy current dampers A torsional device is used as reference being the linear ones

a subset The reference scheme (Kamerbeek, 1973) is a simplified induction motor with one

magnetic pole pair (Figure 3a)

The rotor is made by two windings 1,1’ and 2,2’ installed in orthogonal planes It is crossed

by the constant magnetic field (flux density B ) generated by the stator The analysis is s

performed under the following assumptions:

• the two rotor coils have the same electric parameters and are shorted

• The reluctance of the magnetic circuit is constant The analysis is therefore only

applicable to motional eddy current devices and not to transformer ones (Graves et al.,

2009), (Tonoli et al., 2008)

Fig 2 Sketch of a transformer (a) and a motional damper (b)

Fig 3 a) Sketch of the induction machine b) Mechanical analogue The torque T is balanced

by the force applied to point P by the spring-damper assemblies

• The magnetic flux generated by the stator is constant as if it were produced by

permanent magnets or by current driven electromagnets

• The stator is assumed to be fixed This is equivalent to describe the system in a

reference frame rigidly connected to it

• All quantities are assumed to be independent from the axial coordinate

• Each of the electric parameter is assumed to be lumped

Angle ( )θ t between the plane of winding 2 and the direction of the magnetic field indicates

the angular position of the rotor relative to the stator When currents i r1 and i r2 flow in the

windings, they interact with the magnetic field of the stator and generate a pair of Lorentz

forces (F1,2 in Figure 3a) Each force is perpendicular to both the magnetic field and to the

axis of the conductors They are expressed as:

where N and l r indicate the number of turns in each winding and their axial length The

resulting electromagnetic torques T1 and T2 applied to the rotor of diameter d r are:

1= 1 rsin = rs0sin ,r1 2= 2 rcos = rs0cos r2

where φrs0=Nl d B r r s is the magnetic flux linked with each coil when its normal is aligned

with the magnetic field Bs It represents the maximum magnetic flux The total torque

acting on the rotor is:

Note that the positive orientation of the currents indicated in Figure 3a has been assumed

arbitrarily, the results are not affected by this choice

From the mechanical point of view the eddy current damper behaves then as a crank of

radius φrs0 whose end is connected to two spring/damper series acting along orthogonal

directions Even if the very concept of mechanical analogue is usually a matter of

elementary physics textbooks, the mechanical analogue of a torsional eddy current device is

not common in the literature It has been reported here due to its practical relevance Springs

and viscous dampers can in fact be easily assembled in most mechanical simulation

environments The mechanical analogue in Figure 3b allows to model the effect of the eddy

current damper without needing a multi-domain simulation tool

The model of an eddy current device with p pole pairs can be obtained by considering that

each pair involves two windings electrically excited with 90º phase shift For a one pole pair

device, each pair is associated with a rotor angle of 2π rad; a complete revolution of the

rotor induces one electric excitation cycle of its two windings Similarly, for a p pole pairs

device, each pair is associated to a 2 / pπ rad angle, a complete revolution of the rotor

induces then p excitation cycles on each winding (θe =pθ)

The orthogonality between the two windings allows adopting a complex flux linkage

where j is the imaginary unit Similarly, also the current flowing in the windings can be

written as i r=i r1+ji r2 The total magnetic flux φr linked by each coil is contributed by the

currents i r through the self inductance L r and the flux generated by the stator and linked to

The differential equation governing the complex flux linkage φr is obtained by substituting

eq.(22) in the Kirchoff's voltage law

R L

T p Im e L

θ

The model holds under rather general input angular speed The mechanical torque will be

determined for the following operating conditions:

• coupler: the angular speed is constant: = =θ$ Ω const,

• damper: the rotor is subject to a small amplitude torsional vibration relative to the

jp

φ

The torque (T) to speed (Ω ) characteristic is found by substituting eq.(27) into eq.(26) The

result is the familiar torque to slip speed expression of an induction machine running at

constant speed

2 0

p c

R p

φω

A simple understanding of this characteristic can be obtained by referring to the mechanical

analogue of Figure 3b At speeds such that the excitation frequency is lower than the pole

(pΩ<<ωp), the main contribution to the deformation is that of the dampers, while the

springs behave as rigid bodies The resultant force vector acting on point P is due to the

dampers and acts perpendicularly to the crank φrs0, this produces a counteracting torque

0

=

By converse, at speeds such that pΩ>>ωp the main contribution to the deformation is that

of the springs, while the dampers behave as rigid bodies The resultant force vector on point

P is due to the springs It is oriented along the crank φrs0 and generates a null torque

Damper

If the rotor oscillates (θ( )t =θ0ℜe e( j tω )+ ) with small amplitude about a given angular θm

position θm, the state eq.(24) can be linearized resorting to the small angle assumption

p

j e s

s s

θφφ

ωθ

−

+

where s is the Laplace variable The mechanical impedance Z s m( ), i.e the torque to speed

transfer function is found by substituting eq.(31) into Eq.(26)

This impedance is that of the series connection of a torsional damper and a torsional spring

with viscous damping and spring stiffness given by

that are constant parameters At low frequency (s<<ωp), the device behaves as a pure

viscous damper with coefficient c em This is the term that is taken into account in the

widespread reactive model At high frequency (s>>ωp) it behaves as a mechanical linear

spring with stiffness k em This term on the contrary is commonly neglected in all the models

presented in the literature (Graves et al., 2009), (Nagaya, 1984), (Nagaya & Karube, 1989)

The bandwidth of the mechanical impedance (Figure 4b) is due to the electrical circuit

resistance and inductance It must be taken into account for the design of eddy current

dampers The assumption of neglecting the inductance is valid only for frequency lower

than the electric pole (s<<ωp) The behavior of the mechanical impedance has effects also

on the operation of an eddy current coupler Due to the bandwidth limitations, it behaves as

a low pass filter for each frequency higher than the electric pole

To correlated the torque to speed characteristic of eq.(28) and the mechanical impedance of eq

(32), it shold be analized that the slope c0 of the torque to speed characteristic at zero or low

speed (Ω =pωp) is equal to the mechanical impedance at zero or low frequency (s=ωp):

Fig 4 a) Static characteristic of an axial-symmetric induction machine b) Representation of

its mechanical impedance (magnitude in logarithmic scales)

2 0

em r

0 0

2

rs rs

Fig 5 Sketch of an Active Magnetic Damper in conjunction with a mechanical spring They

both act on the non rotating part of the bearing

A graphical representation of the relationships between eqs.(35) and (36) is given in Figure

4 They allow to obtain the mechanical impedance and/or the state space model valid under

general operating condition, eq.(24), from the torque to speed characteristic This is of

interest because numerical tools performing constant speed analysis are far more common and consolidated than those dealing with transient analysis Vice versa, the steady state torque to speed characteristic can be simply obtained identifying by vibration tests the parameters c em and k em (or ωp)

It's worth to note that eqs.(28), (32) and Eqs.(35), (36) hold in general for eddy current devices with one or more pole pairs They can be applied also to linear electric machies provided that the rotational degree of freedom is transformed into a linear one

2.4 Transformer dampers in active mode (AMD)

Transformer dampers can be used in active mode Active Magnetic Dampers (Figure 5) work in the same way as active magnetic bearings, with the only difference that in this case the force generated by the actuator is not aimed to support the rotor but, in the simplest control strategy, it may be designed just to supply damping; this doesn’t exclude the possibility to develop any more complex control strategy An AMD can be integrated into one of the supports of the rotor In this concept, a rolling element bearing is supported in the housing via mechanical springs providing the required stiffness Both the spring and the damper act on the non-rotating part of the support The stiffness and the load bearing capacity is then provided by the mechanical device while the AMD is used to control the vibrations, adding damping, in its simplest form It is important to note that the stiffness of the springs can be used to compensate the open loop negative stiffness of a typical Maxwell actuator This allows to relieve the active control of the task to guarantee the static stability

of the system A proportional-derivative feedback loop based on the measurement of the support displacements may be enough to control the rotor vibrations Sensors and a controller are then required to this end Under the assumption of typical Maxwell actuators, the force that each coil of the actuator exerts on the moving part is computed by eq.(11), that can be used to design the actuators once its maximum control force is specified It’s worth to note that such damping devices can be applied to any vibrating system

2.5 Transformer dampers in active mode and self-sensing operation

The reversibility of the electromechanical interaction induces an electrical effect when the two parts of an electromagnet are subject to relative motion (back electromotive force) This effect can be exploited to estimate mechanical variables from the measurement of electrical ones This leads to the so-called self-sensing configuration that consists in using the electromagnet either as an actuator and a sensor This configuration permits lower costs and shorter shafts (and thus higher bending frequencies) than classical configurations provided with sensors and non-collocation issues are avoided In practice, voltage and current are used to estimate the airgap To do so, the two main approaches are: the state-space observer approach (Vischer & Bleuler, 1990), (Vischer & Bleuler, 1993) and the airgap estimation using the current ripple (Noh & Maslen, 1997), (Schammass et al., 2005) The former is based

on the electromechanical model of the system As the resulting model is fully observable and controllable, the position and the velocity of the mechanical part can be estimated and fed back to control the vibrations of the system This approach is applicable for voltage-controlled (Mizuno et al., 1996) and current-controlled (Mizuno et al., 1996) electromagnets The second approach takes advantage from the current ripple due to the switching amplifiers to compute in real-time the inductance, and thus the airgap The airgap-estimation can be based on the ripple slope (PWM driven amplifiers, (Okada et al., 1992)) or

on the ripple frequency (hysteresis amplifiers, (Mizuno et al., 1998)) So far in the literature,

self-sensing configurations have been mainly used to achieve the complete suspension of the

rotor The poor robustness of the state-space approach greatly limited its adoption for

industrial applications As a matter of fact, the use of a not well tuned model results in the

system instability (Mizuno et al., 1996) , (Thibeault & Smith 2002) Instead, the direct airgap

estimation approach seems to be more promising in terms of robustness (Maslen et al.,

2006)

Fig 6 Schematic model of electromagnets pair to be used for self-sensing modelling

Here below is described a one degree of freedom mass-spring oscillator actuated by two

opposite electromagnets (Figure 6) Parameters m, k and c are the mass, stiffness and viscous

damping coefficient of the mechanical system The electromagnets are assumed to be

identical, and the coupling between the two electromagnetic circuits is neglected The aim of

the mechanical stiffness is to compensate the negative stiffness due to the electromagnets

Owing to Newton's law in the mechanical domain, the Faraday and Kirchoff laws in the

electrical domain, the dynamics equations of the system are:

where R is the coils resistance and v j is the voltage applied to electromagnet j F d is the

disturbance force applied to the mass, while F1 and F2 are the forces generated by the coils as

in eq (9)

The system dynamics is linearized around a working point corresponding to a bias voltage

v0 imposed to both the electromagnets:

( )

0 0 0

where F0 is the initial force generated by the electromagnets due to the current i0=v0/R,

and Δ is the small variation of the electromagnets' forces As the electromagnets are F j

identical, (i1−i0) (=− −i2 i0)=i c Therefore, a three-state-space model is used to study the

four-state system dynamics described in eq.(37) (Vischer & Bleuler, 1990) The resulting

linearized state-space model is:

where A, B and C are the dynamic, action and output matrices respectively defined as:

with the associated state, input and output vectors X={x x i, ,$ c}T, u={F v d, c}T y=i c

The terms in the matrices derive from the linearization of the non-linear functions defined in

eq (7) and eq (9):

where Γ=μ0N S2 / 2 is the characteristic factor of the electromagnets ,L0, k i, k m and k x

are the inductance, the current-force factor, the back-electromotive force factor, and the

so-called negative stiffness of one electromagnet, respectively The open-loop system is stable

as long as the mechanical stiffness is larger than the total negative stiffness, i.e k+2k x> 0

As eq.(39) describes the open-loop dynamics of the system for small variations of the

variables, and the system stability is insured, the various coefficients of A can be identified

experimentally

Due to the strong nonlinearity of the electromagnetic force as a function of the displacement

and the applied voltage, and to the presence of end stops that limit the travel of the moving

mass, the linear approach may seem to be questionable Nevertheless, the presence of a

mechanical stiffness large enough to overcome the negative stiffness of the electromagnets

makes the linearization point stable, and compels the system to oscillate about it The

selection of a suitable value of the stiffness k is a trade-off issue deriving from the

application requirements However, as far as the linearization is concerned, the larger is the

stiffness k relative to k x , the more negligible the nonlinear effects become

2.5.1 Control design

The aim of the present section is to describe the design strategy of the controller that has

been used to introduce active magnetic damping into the system The control is based on the

Luenberger observer approach (Vischer & Bleuler, 1993), (Mizuno et al., 1996) The adoption

of this approach was motivated by the relatively low level of noise affecting the current

measurement It consists in estimating in real time the unmeasured states (in our case,

displacement and velocity) from the processing of the measurable states (the current) The

observer is based on the linearized model presented previously, and therefore the higher

frequency modes of the mechanical system have not been taken into account Afterwards,

the same model is used for the design of the state-feedback controller

where X∧ and y∧ are the estimations of the system state and output, respectively Matrix L is

commonly referred to as the gain matrix of the observer Eq.(42) shows that the inputs of the

observer are the measurement of the current (y) and the control voltage imposed to the

where = X Xε − Eq (43) emphasizes the role of L in the observer convergence The location ∧

of the eigenvalues of matrix (A LC− ) on the complex plane determines the estimation time

constants of the observer: the deeper they are in the left-half part of the complex plane, the

faster will be the observer It is well known that the observer tuning is a trade-off between

the convergence speed and the noise rejection (Luenberger, 1971) A fast observer is

desirable to increase the frequency bandwidth of the controller action Nevertheless, this

configuration corresponds to high values of L gains, which would result in the amplification

of the unavoidable measurement noise y, and its transmission into the state estimation This

issue is especially relevant when switching amplifiers are used Moreover, the transfer

function that results from a fast observer requires large sampling frequencies, which is not

always compatible with low cost applications

2.5.3 State-feedback controller

A state-feedback control is used to introduce damping into the system The control voltage

is computed as a linear combination of the states estimated by the observer, with K as the

control gain matrix Owing to the separation principle, the state-feedback controller is

designed considering the eigenvalues of matrix (A-BK)

Similarly to the observer, a pole placement technique has been used to compute the gains of

K, so as to maintain the mechanical frequency constant By doing so, the power

consumption for damping is minimized, as the controller does not work against the

mechanical stiffness The idea of the design was to increase damping by shifting the

complex poles closer to the real axis while keeping constant their distance to the origin

(p1 = p2 =constant)

2.6 Semi-active transformer damper

Figure 7 shows the sketch of a “transformer” eddy current damper including two

electromagnets The coils are supplied with a constant voltage and generate the magnetic

field linked to the moving element (anchor) The displacement with speed q$ of the anchor

changes the reluctance of the magnetic circuit and produces a variation of the flux linkage

According to Faraday’s law, the time variation of the flux generates a back electromotive

force Eddy currents are thus generated in the coils The current in the coils is then given by

two contributions: a fixed one due to the voltage supply and a variable one induced by the

back electromotive force The first contribution generates a force that increases with the

decreasing of the air-gap It is then responsible of a negative stiffness The damping force is

generated by the second contribution that acts against the speed of the moving element

Fig 7 Sketch of a two electromagnet Semi Active Magnetic Damper (the elastic support is

omitted)

According to eq (9), considering the two magnetic flux linkages λ1 and λ2 of both

counteracting magnetic circuits, the total force acting on the anchor of the system is:

The state equation relative to the electric circuit can be derived considering a constant

voltage supply common for both the circuits that drive the derivative of the flux leakage and

the voltage drop on the total resistance of each circuit R=R coil +R add (coil resistance and

additional resistance used to tune the electrical circuit pole as:

Where g is the nominal airgap and 0 α=2 /(μ0N A2 )

Eqs.(44) and (45) are linearized for small displacements about the centered position of the

anchor (q= ) to understand the system behavior in terms of poles and zero structure 0

The termλ0=V/(αg R0 ) represents the magnetic flux linkage in the two electromagnets at

steady state in the centered position as obtained from eq.(45) while λ1′ and λ2′ indicate the

variation of the magnetic flux linkages relative to λ0

The transfer function between the speed q$ and the electromagnetic force F shows a first

order dynamic with the pole (ωRL) due to the R-L nature of the circuits

0 0 2

L0 indicates the inductance of each electromagnet at nominal airgap

The mechanical impedance is a band limited negative stiffness This is due to the factor 1/s

and the negative value of K em that is proportional to the electrical power (K m≥ −K em)

dissipated at steady state by the electromagnet

The mechanical impedance and the pole frequency are functions of the voltage supply V

and the resistance R whenever the turns of the windings (N), the air gap area (A) and the

airgap (g0) have been defined The negative stiffness prevents the use of the electromagnet

as support of a mechanical structure unless the excitation voltage is driven by an active

feedback that compensates it This is the principle at the base of active magnetic

suspensions

A very simple alternative to the active feedback is to put a mechanical spring in parallel to

the electromagnet In order to avoid the static instability, the stiffness K m of the added

spring has to be larger than the negative electromechanical stiffness of the damper

(K m≥ −K em) The mechanical stiffness could be that of the structure in the case of an already

supported structure Alternatively, if the structure is supported by the dampers themselves,

the springs have to be installed in parallel to them As a matter of fact, the mechanical spring

in parallel to the transformer damper can be considered as part of the damper